Simplify the following expression and state the condition under which the simplification is valid: $z = \dfrac{r^2 + 6r}{r^2 + 9r + 18}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{r^2 + 6r}{r^2 + 9r + 18} = \dfrac{(r)(r + 6)}{(r + 3)(r + 6)} $ Notice that the term $(r + 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(r + 6)$ gives: $z = \dfrac{r}{r + 3}$ Since we divided by $(r + 6)$, $r \neq -6$. $z = \dfrac{r}{r + 3}; \space r \neq -6$